Improved Ordering Results for Fail-Safe Systems with Exponential Components

LetX(2:n) and Y-2:m be the second order statistics from n independent exponential variables with hazards lambda(1), ... , lambda(n), and an independent exponential sample of size m with hazard change to lambda, respectively. When m >= n, we obtain necessary and sufficient conditions for comparing X-2:n and Y-2:m in mean residual life, dispersive, hazard rate, and likelihood ratio orderings based on some inequalities between lambda(i)'s and lambda. The established results show how one can compare an (n - 1)-out-of-n system consisting of heterogeneous components with exponential lifetimes with any (m - 1)-out-of-m system consisting of homogeneous components with exponential lifetimes.